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Luis Mantejano

Corollary 3 follows directly from Theorem 1 when N = A and

h = Id.

In the same way that the Chapman-Ferry Theorem has the Siebermann

Cell-like Approximation Theorem[21] as a corollary, we have as a

consequence of Corollary 3 that codimension three cell-like maps

between manifolds have locally flat approximate cross sections, thus

solving a conjecture posed by Rushing in [20].

In codimension two, we also study the general lifting-problem

dDtaining similar conslusions provided that certain added conditions

are satisfied. In particular we apply Venema's approximation

techniques, [22] and [23] to obtain the following two theorems:

1HEOREM 2. Let A be an ANR , h: D + A an embedding and

U an open subset of D . Given e 0 there exists 6 0 such

that if is a 6-fibration from a PL manifold

onto A such that there exists a locally flat PL embedding

g : U • * M with d(pgn,h|u) 6, then there exists a locally flat

PL embedding g: D • M with d(pg,h) e.

THEOREM 3. Let w be a coitpact connected 2-manifold with

boundary, A an ANR and h: N -* • A an embedding. Given e 0

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there exists 6 0 such that if p: M • * A is a 6-homotopy equiva-

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lence from a PL 4-manifold M onto A, then there exists a local-

ly flat PL embedding g: w • * M such that d(pg,h) e.

Similar corollaries to those obtained from Theorem 1 can be

obtained from Theorem 2 and Theorem 3.

Results of Chapman [7] and Edwards [11] allow us to obtain the